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**Non-negative pinching, moduli spaces and bundles with infinitely many souls.**
*(English)*
Zbl 1102.53020

The authors discuss several infiniteness phenomena in non-negative sectional curvature. The work is motivated by the finiteness theorems in Riemannian geometry and a question of S.-T. Yau which asks whether there always exists only a finite number of diffeomorphism types of closed smooth manifolds of positive sectional curvature that are homotopy equivalent to a given positively curved manifold [Proc. Symp. Pure Math. 54, Part 1, 1–28 (1993; Zbl 0801.53001)]. In connection with Yau’s problem, the following two questions arise:

1) Given fixed \(n \in \mathbb{N}, D>0\) and \(c, C \in \mathbb{R}\), are there at most finitely many diffeomorphism classes of pairwise homotopy equivalent closed Riemannian \(n\)-manifolds \(M^{n}\) with sectional curvature \(c \leq \text{sec} \leq C\) and diameter \(\leq D\)?

2) Given fixed \(n \in \mathbb{N}\) and \(C, D>0\), are there always at most finitely many diffeomorphism types of pairwise homotopy equivalent closed Riemannian \(n\)-manifolds with sectional curvature \(0\leq \text{sec} \leq C\) and diameter \(\leq D\)?

The authors give some answers to these questions. In this article it is shown that in each dimension \(n\geq 10\), there exist infinite sequences of homotopy equivalent, but mutually non-homeomorphic closed simply connected Riemannian \(n\)-manifolds with \(0 \leq \text{sec} \leq 1\), positive Ricci curvature and uniformly bounded diameter. The authors have constructed examples of manifolds admitting infinitely many non-negatively curved metrics with mutually non-homeomorphic souls which have uniform bounds on the curvature of the manifolds and the diameters of the souls. Examples of non-compact manifolds whose moduli spaces of complete metrics with \(\text{sec} \geq 0\) have infinitely many connected components are constructed, too.

1) Given fixed \(n \in \mathbb{N}, D>0\) and \(c, C \in \mathbb{R}\), are there at most finitely many diffeomorphism classes of pairwise homotopy equivalent closed Riemannian \(n\)-manifolds \(M^{n}\) with sectional curvature \(c \leq \text{sec} \leq C\) and diameter \(\leq D\)?

2) Given fixed \(n \in \mathbb{N}\) and \(C, D>0\), are there always at most finitely many diffeomorphism types of pairwise homotopy equivalent closed Riemannian \(n\)-manifolds with sectional curvature \(0\leq \text{sec} \leq C\) and diameter \(\leq D\)?

The authors give some answers to these questions. In this article it is shown that in each dimension \(n\geq 10\), there exist infinite sequences of homotopy equivalent, but mutually non-homeomorphic closed simply connected Riemannian \(n\)-manifolds with \(0 \leq \text{sec} \leq 1\), positive Ricci curvature and uniformly bounded diameter. The authors have constructed examples of manifolds admitting infinitely many non-negatively curved metrics with mutually non-homeomorphic souls which have uniform bounds on the curvature of the manifolds and the diameters of the souls. Examples of non-compact manifolds whose moduli spaces of complete metrics with \(\text{sec} \geq 0\) have infinitely many connected components are constructed, too.

Reviewer: Ilie Burdujan (Iaşi)

### MSC:

53C20 | Global Riemannian geometry, including pinching |

58D17 | Manifolds of metrics (especially Riemannian) |